Asymptotic theory for local estimators based on Bregman divergence

This article is concerned with asymptotic theory for local estimators based on Bregman divergence. We consider a localized version of Bregman divergence induced by a kernel weight and minimize it to obtain the local estimator. We provide a rigorous proof for the asymptotic consistency of the local estimator in a situation where both the sample size and the bandwidth involved in the kernel weight increase. Asymptotic normality of the local estimator is also developed under the same asymptotic scenario. Monte Carlo simulations are also performed to confirm the theoretical results. The Canadian Journal of Statistics 47: 628–652; 2019 © 2019 Statistical Society of Canada     

DASSO: connections between the Dantzig selector and lasso

Summary.  We propose a new algorithm, DASSO, for fitting the entire coefficient path of the Dantzig selector with a similar computational cost to the least angle regression algorithm that is used to compute the lasso. DASSO efficiently constructs a piecewise linear path through a sequential simplex-like algorithm, which is remarkably similar to the least angle regression algorithm. Comparison of the two algorithms sheds new light on the question of how the lasso and Dantzig selector are related. In addition, we provide theoretical conditions on the design matrix X under which the lasso and Dantzig selector coefficient estimates will be identical for certain tuning parameters. As a consequence, in many instances, we can extend the powerful non-asymptotic bounds that have been developed for the Dantzig selector to the lasso. Finally, through empirical studies of simulated and real world data sets we show that in practice, when the bounds hold for the Dantzig selector, they almost always also hold for the lasso.     

Automatic bandwidth selection for modified m-smoother ∗

Traditional parametric and nonparametric regression techniques encounter serious over smoothing problems when jump point discontinuities exist in the underlying mean function. Recently, Chu, Glad, Godtliebsen and Marron (1998) developed a method using a modified M-smoothing technique to preserve jumps and spikes while producing a smooth estimate of the mean function. The performance of Chu etal.'s (1998) method is quite sensitive to the choice of the required bandwidths g and h. Furthermore, it is not obvious how to extend certain commonly used automatic bandwidth selection procedures when jumps and spikes are present. In this paper we propose a rule of thumb method of choosing the smoothing parameters based on asymptotic optimal bandwidth formulas and robust estimates of unknown quantities. We also evaluate the proposed bandwidth selection method via a small simulation study.     

A simple bootstrap bandwidth selector for local polynomial fitting

A new, fully data-driven bandwidth selector with a double smoothing (DS) bias term and a data-driven variance estimator is developed following the bootstrap idea. The data-driven variance estimation does not involve any additional bandwidth selection. The proposed bandwidth selector convergences faster than a plug-in one due to the DS bias estimate, whereas the data-driven variance improves its finite sample performance clearly and makes it stable. Asymptotic results of the proposals are obtained. A comparative simulation study was done to show the overall gains and the gains obtained by improving either the bias term or the variance estimate, respectively. It is shown that the use of a good variance estimator is more important when the sample size is relatively small.     

Local linear variable bandwidth hazard rate estimation

A new hazard rate estimator under the random right censorship model is proposed in this article. The estimator arises naturally as a combination of the local linear fitting and variable bandwidth methods. As a consequence, it also inherits the benefits of both approaches. The asymptotic properties of the estimate in the boundary and in the interior of the region of estimation are provided and its asymptotic distribution is established. In addition, an automatic data-driven bandwidth selection procedure is proposed and evaluated via Monte Carlo simulations. Further numerical studies compare the performance of the proposed estimate with that of estimates with similar asymptotic properties.     

Kernel estimation of regression function gradient

Abstract

This paper is focused on kernel estimation of the gradient of a multivariate regression function. Despite the importance of this topic, the progress in this area is rather slow. Our aim is to construct a gradient estimator using the idea of local linear estimator for a regression function. The quality of this estimator is expressed in terms of the Mean Integrated Square Error. We focus on a choice of bandwidth matrix. Further, we present some data-driven methods for its choice and develop a new approach. The performance of presented methods is illustrated using a simulation study and real data example.     

Further theoretical and practical insight to the do-validated bandwidth selector

Recent contributions to kernel smoothing show that the performance of cross-validated bandwidth selectors improves significantly from indirectness and that the recent do-validated method seems to provide the most practical alternative among these methods. In this paper we show step by step how classical cross-validation improves in theory, as well as in practice, from indirectness and that do-validated estimators improve in theory, but not in practice, from further indirectness. This paper therefore provides a strong support for the practical and theoretical properties of do-validated bandwidth selection. Do-validation is currently being introduced to survival analysis in a number of contexts and this paper provides evidence that this might be the immediate step forward.     

Automatic Local Smoothing for Spectral Density Estimation

This article uses local polynomial techniques to fit Whittle's likelihood for spectral density estimation. Asymptotic sampling properties of the proposed estimators are derived, and adaptation of the proposed estimator to the boundary effect is demonstrated. We show that the Whittle likelihood-based estimator has advantages over the least-squares based log-periodogram. The bandwidth for the Whittle likelihood-based method is chosen by a simple adjustment of a bandwidth selector proposed in Fan & Gijbels (1995). The effectiveness of the proposed procedure is demonstrated by a few simulated and real numerical examples. Our simulation results support the asymptotic theory that the likelihood based spectral density and log-spectral density estimators are the most appealing among their peers     

An iterative plug-in algorithm for decomposing seasonal time series using the Berlin Method

We propose a fast data-driven procedure for decomposing seasonal time series using the Berlin Method, the procedure used, e.g. by the German Federal Statistical Office in this context. The formula of the asymptotic optimal bandwidth h _A is obtained. Methods for estimating the unknowns in h _A are proposed. The algorithm is developed by adapting the well-known iterative plug-in idea to time series decomposition. Asymptotic behaviour of the proposal is investigated. Some computational aspects are discussed in detail. Data examples show that the proposal works very well in practice and that data-driven bandwidth selection offers new possibilities to improve the Berlin Method. Deep insights into the iterative plug-in rule are also provided.     

A new non parametric estimator for Pdf based on inverse gamma distribution

ABSTRACT

The non parametric approach is considered to estimate probability density function (Pdf) which is supported on(0, ∞). This approach is the inverse gamma kernel. We show that it has same properties as gamma, reciprocal inverse Gaussian, and inverse Gaussian kernels such that it is free of the boundary bias, non negative, and it achieves the optimal rate of convergence for the mean integrated squared error. Also some properties of the estimator were established such as bias and variance. Comparison of the bandwidth selection methods for inverse gamma kernel estimation of Pdf is done.